Model Rainfall Extremes on an Australian Wide Scale
Aim
Challenge
Clustering
Distance for clustering is based on extremal dependence.
Calculate our distance \(d(x, x+h)\) using the F-madogram (Cooley 2006):
\[ d(x, x+h) = \dfrac{1}{2} \mathbb{E} \left| \left[ F(Z(x)) - F(Z(x+h)) \right| \right] \] where
\(F\) is the empirical distribution function of
\(Z(x)\) the annual maximum rainfall at location
\(x\) .
Pros: Non-parametric, fast and only use the raw data.
Can write our F-madogram as a function of the extremal coefficient, \(\theta(h)\) \[ d(x,x + h) = \dfrac{\theta(h) - 1}{2(\theta(h) + 1)}, \]
where \(\theta(h)\) is a measure of extremal dependence,
\[ \mathbb{P}\left( Z(x) \leq z, Z(x + h) \leq z \right) = \left[\mathbb{P}(Z(x)\leq z) \right]^{\theta(h)}, \quad x,h \in \mathbb{R}^2 \] where
\(Z\) is unit Frechet.
(For full details refer to Bernard et al. 2013.)
Clustering - K-medoids
Model
Local level:
+ Fit a Max-stable process
+ Natural extension of univariate extreme value theory to extremes in continuous space with dependence
Classification - KNN
Questions?
This work has been supported by the Australian Research Council through the Laureate Fellowship FL130100039, CSIRO and ACEMS.
